Value functions and the Hamilton--Jacobi--Bellman equation: orientation only

For a controlled dynamical system $y^{\mathrm{\prime }}(\cdot )=b(y,v)$ with running cost $L(y,v)$ and initial cost $u_0$, consider the value function $u(x,t)=\inf \left\{{\int }_0^{t}L\right(\gamma (s),v(s))ds+u_0(\gamma (0)):\gamma (t)=x,{\gamma }^{\mathrm{\prime }}=b(\gamma ,v)\}$ over admissible controls. When hypotheses make this value finite and continuous, ensure the dynamic programming principle, and give the viscosity characterization, $u$ is a viscosity solution of the Hamilton--Jacobi--Bellman equation $u_t+H(x,Du)=0$ with $H(x,p)={\sup }_v\{p\cdot b(x,v)-L(x,v)\}\mathrm{.}$ To express the Legendre duality precisely, define the effective velocity cost $\mathrm{\ell }(x,\xi ):=\inf \{L(x,v):b(x,v)=\xi \},$ with value $+\mathrm{\infty }$ when the fiber is empty. Then $H(x,\cdot )$ is the convex conjugate of $\mathrm{\ell }(x,\cdot )$; when $\mathrm{\ell }(x,\cdot )$ is proper, lower semicontinuous and convex, Fenchel--Moreau gives $\mathrm{\ell }(x,\xi )={\sup }_p\{p\cdot \xi -H(x,p)\}$ (Clason, Theorem 5.1(iii), recorded here as source-only orientation; The Legendre transform of a finite-valued convex Hamiltonian fixes the conjugate notation but does not prove this extended-valued result). If comparison holds in the chosen solution class, the viscosity solution is unique.

The sign convention is the one of this page: velocities are integrated forward from time $0$ to time $t$, the running cost is accumulated forward, the initial cost is paid at time $0$, and $H$ is convex in $p$ because it is a supremum of affine functions of $p$, irrespective of convexity of the control or velocity set. With this convention the Hopf--Lax operator of The Hopf--Lax operator and the Hopf--Lax formula is the special case of the formula in which the infimum over paths has been reduced to a single infimum over the starting point, $u(x,t)={\inf }_y\{u_0(y)+tL((x-y)\mathrm{/}t)\}$, for the autonomous convex superlinear case.

This remark records the interpretation only: no admissible-control framework, no measurable selection, no existence of optimal controls and no dynamic programming theorem for control systems is developed on this page, The stationary specialization of the displayed evolution equation is $H(x,Du)=0$; a separately discounted formulation leads to equations such as $\lambda u+H(x,Du)=f$. The only dynamic-programming content of this page is the semigroup law for the Hopf--Lax operator The Hopf--Lax operators form a semigroup (dynamic programming), which is proved directly from the convexity of the Lagrangian. No choice principle is used; this orientation is recorded so that the Cauchy problems of The Hamilton--Jacobi Cauchy problem and its classical solutions can be read against their control origin.